Modified Scattering and Asymptotics for Perturbed One-Dimensional Cubic NLS
Abstract
We study the long-time dynamics of small solutions to the one-dimensional nonlinear Schrödinger equation i∂t v+∂x2v−βabsv2v +cW(x)absv4v+γi∂xv=0, where cW is spatially localized. The cubic nonlinearity is long range and produces the logarithmic phase correction, whereas the localized quintic term is short range at leading order. We construct a global forward modified wave operator for small complex asymptotic profiles and prove quantitative final-state estimates. For small data in the weighted energy space, we also establish global existence, sharp t−1/2 decay, and forward modified scattering with a unique asymptotic profile. The principal new phenomenon occurs beyond this leading law. The exact Duhamel tail generated by the localized quintic term admits a quantitative inner scaling limit at the distinguished frequency ζ=−γ/2 on the scale absζ+γ/2∼ t−1/2. Its universal shape is explicit and depends on the value of the scattering profile on the distinguished ray and on the zeroth moment of cW. When both quantities are nonzero, the limit is nontrivial, belongs optimally to C2,1loc, and is not C3 at the center. Away from the corresponding self-similar ray ξ=−γ, we construct rigorously defined higher-order outer expansions through every integer order allowed by the decay of cW, and to each fixed finite order when cW is rapidly decreasing.
Related
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BibTeX
@misc{nguyen2026modified,
title = {{Modified Scattering and Asymptotics for Perturbed One-Dimensional Cubic NLS}},
author = {Vinh Nguyen and Avy Soffer and Truong Vu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07044},
eprint = {2609.07044},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07044},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Modified Scattering and Asymptotics for Perturbed One-Dimensional Cubic NLS
AU - Vinh Nguyen
AU - Avy Soffer
AU - Truong Vu
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07044
UR - https://arxiv.org/abs/2609.07044
AB - We study the long-time dynamics of small solutions to the one-dimensional nonlinear Schrödinger equation i∂_t v+∂_x^2v−βabsv^2v +cW(x)absv^4v+γi∂_xv=0, where cW is spatially localized. The cubic nonlinearity is long range and produces the logarithmic phase correction, whereas the localized quintic term is short range at leading order. We construct a global forward modified wave operator for small complex asymptotic profiles and prove quantitative final-state estimates. For small data in the weighted energy space, we also establish global existence, sharp t^−1/2 decay, and forward modified scattering with a unique asymptotic profile. The principal new phenomenon occurs beyond this leading law. The exact Duhamel tail generated by the localized quintic term admits a quantitative inner scaling limit at the distinguished frequency ζ=−γ/2 on the scale absζ+γ/2∼ t^−1/2. Its universal shape is explicit and depends on the value of the scattering profile on the distinguished ray and on the zeroth moment of cW. When both quantities are nonzero, the limit is nontrivial, belongs optimally to C^2,1_loc, and is not C^3 at the center. Away from the corresponding self-similar ray ξ=−γ, we construct rigorously defined higher-order outer expansions through every integer order allowed by the decay of cW, and to each fixed finite order when cW is rapidly decreasing.
ER -